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1、三重積分的概念與計(jì)算2013.4.15 with(plots):a:=(sqrt(5)-1)/2: u:=(x,y)-sqrt(x2+y2): v:=(x,y)-1-x2-y2:qumian1:=plot3d(u(x,y),x=-a.a,y=-sqrt(a2-x2).sqrt(a2-x2),color=yellow):qumian2:=plot3d(v(x,y),x=-a.a,y=-sqrt(a2-x2).sqrt(a2-x2),color=green):quxian:=spacecurve(a*cos(t),a*sin(t),u(a*cos(t),a*sin(t),t=0.2*Pi,colo

2、r=red,thickness=4): x_axis:=plot3d(u,0,0,u=-1/2.1/2,v=0.0.01,thickness=3):y_axis:=plot3d(0,u,0,u=-1/2.1/2,v=0.0.01,thickness=3):z_axis:=plot3d(0,0,u,u=-.2.1.2,v=0.0.01,thickness=3):xyz:=display(x_axis,y_axis,z_axis,thickness=3):display(qumian1,qumian2,quxian,xyz,scaling=constrained,orientation=20,60

3、); with(plots):R:=(sqrt(5)-1)/2:qumian1:=plot3d(s*cos(t),s*sin(t),s,t=0.2*Pi,s=0.R,color=yellow):qumian2:=plot3d(s*cos(t),s*sin(t),1-s2,t=0.2*Pi,s=0.R,color=green):x_axis:=plot3d(u,0,0,u=-1/2.1/2,v=0.0.01,thickness=3):y_axis:=plot3d(0,u,0,u=-1/2.1/2,v=0.0.01,thickness=3):z_axis:=plot3d(0,0,u,u=-.2.1

4、.2,v=0.0.01,thickness=3):xyz:=display(x_axis,y_axis,z_axis,thickness=3):display(qumian1,qumian2,xyz,scaling=constrained,orientation=33,68) f:=(x,y,z)-y*cos(x+z):x1:=0:x2:=Pi/2:y1:=x-0:y2:=x-sqrt(x):z1:=(x,y)-0:z2:=(x,y)-Pi/2-x:Int(Int(Int(f(x,y,z),z=z1(z,y).z2(z,y),y=y1(x).y2(x),x=x1.x2)=int(int(int

5、(f(x,y,z),z=z1(x,y).z2(x,y),y=y1(x).y2(x),x=x1.x2); d02 d0 x d0 2 zy ( )cos x z z y x 216 12 with(plots):qumian1:=plot3d(x,0,z,x=0.Pi/2,z=0.Pi/2-x,color=yellow):qumian2:=plot3d(x,y,0,x=0.Pi/2,y=0.sqrt(x),color=green):qumian3:=plot3d(x,sqrt(x),z,x=0.Pi/2,z=0.Pi/2-x,color=brown):qumian4:=plot3d(x,y,Pi

6、/2-x,x=0.Pi/2,y=0.sqrt(x),color=gray):x_axis:=plot3d(u,0,0,u=-1/2.2,v=0.0.01,thickness=3):y_axis:=plot3d(0,u,0,u=-1/2.1,v=0.0.01,thickness=3):z_axis:=plot3d(0,0,u,u=-.2.2,v=0.0.01,thickness=3):xyz:=display(x_axis,y_axis,z_axis,thickness=3):display(qumian1,qumian2,qumian3,qumian4,xyz,scaling=constrai

7、ned,orientation=55,68); f:=(x,y,z)-x*z/(1+y)2:x1:=0:x2:=1:y1:=x-x2:y2:=x-1:z1:=(x,y)-0:z2:=(x,y)-1-y2:Int(Int(Int(f(x,y,z),z=z1(z,y).z2(z,y),y=y1(x).y2(x),x=x1.x2)=int(int(int(f(x,y,z),z=z1(x,y).z2(x,y),y=y1(x).y2(x),x=x1.x2); d01 dx21 d0 1 y2 x z( )1 y 2 z y x 148 with(plots):qumian1:=plot3d(0,y,z,

8、y=0.1,z=0.1-y2,color=yellow):qumian2:=plot3d(x,y,0,x=0.1,y=x2.1,color=green):qumian3:=plot3d(x,x2,z,x=0.1,z=0.1-x4,color=brown):qumian4:=plot3d(x,y,1-y2,x=0.1,y=x2.1,color=gray):x_axis:=plot3d(u,0,0,u=-1/2.1,v=0.0.01,thickness=3):y_axis:=plot3d(0,u,0,u=-1/2.1.5,v=0.0.01,thickness=3):z_axis:=plot3d(0

9、,0,u,u=-.2.1.5,v=0.0.01,thickness=3):xyz:=display(x_axis,y_axis,z_axis,thickness=3):display(qumian1,qumian2,qumian3,qumian4,xyz,scaling=constrained,orientation=55,68); f:=(x,y,z)-exp(abs(z);theta1:=0:theta2:=2*Pi:r1:=theta-0:r2:=theta-sqrt(1-z2):z1:=(x,y)-1:z2:=(x,y)-1:Int(Int(Int(f(x,y,z)*r,r=r1(th

10、eta).r2(theta),theta=theta1.theta2),z=z1.z2)=int(int(int(f(x,y,z)*r,r=r1(theta).r2(theta),theta=theta1.theta2),z=z1.z2); d-11 d02 d0 1 z2e z r r z 2 f:=(x,y,z)-x:x1:=0:x2:=1:y1:=x-0:y2:=x-(1-x)/(1+x):z1:=(x,y)-0:z2:=(x,y)-x*y:Int(Int(Int(f(x,y,z),z=z1(z,y).z2(z,y),y=y1(x).y2(x),x=x1.x2)=int(int(int(

11、f(x,y,z),z=z1(x,y).z2(x,y),y=y1(x).y2(x),x=x1.x2); d01 d0 1 xx 1 d0y zx z y x 256 6 ( )ln 2f:=(x,y,z)-x:x1:=0:x2:=1:y1:=x-(1-x)/(1+x):y2:=x-1-x:z1:=(x,y)-0:z2:=(x,y)-1-x-y:Int(Int(Int(f(x,y,z),z=z1(z,y).z2(z,y),y=y1(x).y2(x),x=x1.x2)=int(int(int(f(x,y,z),z=z1(x,y).z2(x,y),y=y1(x).y2(x),x=x1.x2); d01

12、 d 1 x1 x1 x d0 1 z yx z y x 13324 8 ( )ln 2 with(plots):pingmian1:=implicitplot3d(z=0,x=-.2.1.2,y=-0.2.1.2,z=-1.1,color=grey,style=patchnogrid):pingmian2:=implicitplot3d(z=1-x-y,x=-.2.1.2,y=-0.2.1.2,z=-0.5.1.2,color=green,style=patchnogrid):qumian:=implicitplot3d(z=x*y,x=-.2.1.2,y=-0.2.1.2,z=-1.1.3

13、,color=brown,style=patch):x_axis:=plot3d(u,0,0,u=-1.1.3,v=0.0.01,thickness=3):y_axis:=plot3d(0,u,0,u=-1.1.5,v=0.0.01,thickness=3):z_axis:=plot3d(0,0,u,u=-.3.1.5,v=0.0.01,thickness=3):xyz:=display(x_axis,y_axis,z_axis,thickness=3): text:=textplot3d(0,0,-.8,SCUMATH 2008,color=grey):display(qumian,ping

14、mian1,pingmian2,text,xyz,scaling=constrained); 0, 1 ,z z x y z xy 以下曲面所圍成的立體： 1z xyz x y 1xy x y 11 xy x 1D 2D在D1上：0 z xy 在D2上：0 1z x y with(plots):pingmian1:=plot3d(x,y,0,x=0.1,y=0.1-x,color=grey,style=patchnogrid):pingmian2:=plot3d(x,y,1-x-y,x=0.1,y=(1-x)/(1+x).1-x,color=green,style=patchnogrid):q

15、umian:=plot3d(x,y,x*y,x=0.1,y=0.(1-x)/(1+x),color=brown,style=patchnogrid):x_axis:=plot3d(u,0,0,u=-.1.1.1,v=0.0.01,thickness=3):y_axis:=plot3d(0,u,0,u=-.1.1.1,v=0.0.01,thickness=3):z_axis:=plot3d(0,0,u,u=-.1.0.2,v=0.0.01,thickness=3):xyz:=display(x_axis,y_axis,z_axis,thickness=3):display(qumian,ping

16、mian1,pingmian2,xyz,scaling=constrained); with(plots):pingmian:=plot3d(x,y,0,x=0.1,y=0.1-x,color=grey,style=patchnogrid):qumian2:=plot3d(x,(1-x)/(1+x),z,x=0.1,z=0.x*(1-x)/(1+x),color=gold,style=patchnogrid):qumian:=plot3d(x,y,x*y,x=0.1,y=0.(1-x)/(1+x),color=brown,style=patchnogrid):x_axis:=plot3d(u,0,0,u=-.1.1.1,v=0.0.01,thickness=3):y_axis:=plot3d(0,u,0,u=-.1.1.1,v=0.0.01,thickness=3):z_axis:=plot3d(0,0,u,u=-.1.0.2,v=0.0.01,thickness=3):xyz:=display(x_axis,y_axis,z_axis,thickness=3):display(qumian,pingmian,qumian2,xyz,scaling=unconstrained);